On the origin of the cooperativity of protein folding: implications from model simulations

The role of local versus nonlocal physicochemical restraints in determining protein native structure

The tertiary structure of a native protein is dictated by the interplay of local secondary structure propensities, hydrogen bonding, and tertiary interactions. It is argued that the space of known protein topologies covers all single domain folds and results from the compactness of the native structure and excluded volume. Protein compactness combined with the chirality of the protein's side chains also yields native-like Ramachandran plots. It is the many-body, tertiary interactions among residues that collectively select for the global structure that a particular protein sequence adopts. This explains why the recent advances in deep-learning approaches that predict protein side-chain contacts, the distance matrix between residues, and sequence alignments are successful. They succeed because they implicitly learned the many-body interactions among protein residues.

Protein structure predictions by enhanced conformational sampling methods

In this Special Festschrift Issue for the celebration of Professor Nobuhiro Gō's 80th birthday, we review enhanced conformational sampling methods for protein structure predictions. We present several generalized-ensemble algorithms such as multicanonical algorithm, replica-exchange method, etc. and parallel Monte Carlo or molecular dynamics method with genetic crossover. Examples of the results of these methods applied to the predictions of protein tertiary structures are also presented.

Concerted millisecond timescale dynamics in the intrinsically disordered carboxyl terminus of γ-tubulin induced by mutation of a conserved tyrosine residue

Tubulins are an ancient family of eukaryotic proteins characterized by an amino-terminal globular domain and disordered carboxyl terminus. These carboxyl termini play important roles in modulating the behavior of microtubules in living cells. However, the atomic-level basis of their function is not well understood. These regions contain multiple acidic residues and their overall charges are modulated in vivo by post-translational modifications, for example, phosphorylation. In this study, we describe an application of NMR and computer Monte Carlo simulations to investigate how the modification of local charge alters the conformational sampling of the γ-tubulin carboxyl terminus. We compared the dynamics of two 39-residue polypeptides corresponding to the carboxyl-terminus of yeast γ-tubulin. One polypeptide comprised the wild-type amino acid sequence while the second contained a Y > D mutation at Y11 in the polypeptide (Y445 in the full protein). This mutation introduces additional negative charge at a site that is phosphorylated in vivo and produces a phenotype with perturbed microtubule function. NMR relaxation measurements show that the Y11D mutation produces dramatic changes in the millisecond-timescale motions of the entire polypeptide. This observation is supported by Monte Carlo simulations that-similar to NMR-predict the WT γ-CT is largely unstructured and that the substitution of Tyr 11 with Asp causes the sampling of extended conformations that are unique to the Y11D polypeptide.

Design of a rotamer library for coarse-grained models in protein-folding simulations

Rotamer libraries usually contain geometric information to trace an amino acid side chain, atom by atom, onto a protein backbone. These libraries have been widely used in protein design, structure refinement and prediction, homology modeling, and X-ray and NMR structure validation. However, they usually present too much information and are not always fully compatible with the coarse-grained models of the protein geometry that are frequently used to tackle the protein-folding problem through molecular simulation. In this work, we introduce a new backbone-dependent rotamer library for side chains compatible with low-resolution models in polypeptide chains. We have dispensed with an atomic description of proteins, representing each amino acid side chain by its geometric center (or centroid). The resulting rotamers have been estimated from a statistical analysis of a large structural database consisting of high-resolution X-ray protein structures. As additional information, each rotamer includes the frequency with which it has been found during the statistical analysis. More importantly, the library has been designed with a careful control to ensure that the vast majority of side chains in protein structures (at least 95% of residues) are properly represented. We have tested our library using an independent set of proteins, and our results support a good correlation between the reconstructed centroids from our rotamer library and those in the experimental structures. This new library can serve to improve the definition of side chain centroids in coarse-grained models, avoiding at the same time an excessive additional complexity in a geometric model for the polypeptide chain.

Enhanced sampling algorithms

In biomolecular systems (especially all-atom models) with many degrees of freedom such as proteins and nucleic acids, there exist an astronomically large number of local-minimum-energy states. Conventional simulations in the canonical ensemble are of little use, because they tend to get trapped in states of these energy local minima. Enhanced conformational sampling techniques are thus in great demand. A simulation in generalized ensemble performs a random walk in potential energy space and can overcome this difficulty. From only one simulation run, one can obtain canonical-ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review uses of the generalized-ensemble algorithms in biomolecular systems. Three well-known methods, namely, multicanonical algorithm, simulated tempering, and replica-exchange method, are described first. Both Monte Carlo and molecular dynamics versions of the algorithms are given. We then present various extensions of these three generalized-ensemble algorithms. The effectiveness of the methods is tested with short peptide and protein systems.

Comparative modeling and protein-like features of hydrophobic-polar models on a two-dimensional lattice

Lattice models of proteins have been extensively used to study protein thermodynamics, folding dynamics, and evolution. Our study considers two different hydrophobic-polar (HP) models on the 2D square lattice: the purely HP model and a model where a compactness-favoring term is added. We exhaustively enumerate all the possible structures in our models and perform the study of their corresponding folds, HP arrangements in space and shapes. The two models considered differ greatly in their numbers of structures, folds, arrangements, and shapes. Despite their differences, both lattice models have distinctive protein-like features: (1) Shapes are compact in both models, especially when a compactness-favoring energy term is added. (2) The residue composition is independent of the chain length and is very close to 50% hydrophobic in both models, as we observe in real proteins. (3) Comparative modeling works well in both models, particularly in the more compact one. The fact that our models show protein-like features suggests that lattice models incorporate the fundamental physical principles of proteins. Our study supports the use of lattice models to study questions about proteins that require exactness and extensive calculations, such as protein design and evolution, which are often too complex and computationally demanding to be addressed with more detailed models.

FOLD HELICAL PROTEINS BY ENERGY MINIMIZATION IN DIHEDRAL SPACE AND A DFIRE-BASED STATISTICAL ENERGY FUNCTION

Statistical energy functions are discrete (or stepwise) energy functions that lack van der Waals repulsion. As a result, they are often applied directly to a given structure (native or decoy) without further energy minimization being performed to the structure. However, the full benefit (or hidden defect) of an energy function cannot be revealed without energy minimization. This paper tests a recently developed, all-atom statistical energy function by energy minimization with a fixed secondary helical structure in dihedral space. This is accomplished by combining the statistical energy function based on a distance-scaled finite ideal-gas reference (DFIRE) state with a simple repulsive interaction and an improper torsion energy function. The energy function was used to minimize 2000 random initial structures of 41 small and medium-sized helical proteins in a dihedral space with a fixed helical region. Results indicate that near-native structures for most studied proteins can be obtained by minimization alone. The average minimum root-mean-squared distance (rmsd) from the native structure for all 41 proteins is 4.1 Å. The energy function (together with a simple clustering of similar structures) also makes a reasonable selection of near-native structures from minimized structures. The average rmsd value and the average rank for the best structure in the top five is 6.8 Å and 2.4, respectively. The accuracy of the structures sampled and the structure selections can be improved significantly with the removal of flexible terminal regions in rmsd calculations and in minimization and with the increase in the number of minimizations. The minimized structures form an excellent decoy set for testing other energy functions because most structures are well-packed with minimum hard-core overlaps with correct hydrophobic/hydrophilic partitioning. They are available online at .

The folding energy landscape and free energy excitations of cytochrome c

The covalently bound heme cofactor plays a dominant role in the folding of cytochrome c. Because of the complicated inorganic chemistry of the heme, some might consider the folding of cytochrome c to be a special case, following principles different from those used to describe the folding of proteins without cofactors. Recent investigations, however, demonstrate that common models describing folding for many proteins work well for cytochrome c when heme is explicitly introduced, generally providing results that agree with experimental observations. In this Account, we first discuss results from simple native structure-based models. These models include attractive interactions between nonadjacent residues only if they are present in the crystal structure at pH 7. Because attractive nonnative contacts are not included in native structure-based models, their energy landscapes can be described as "perfectly funneled". In other words, native structure-based models are energetically guided towards the native state and contain no energetic traps that would hinder folding. Energetic traps are denoted sources of "frustration", which cause specific transient intermediates to be populated. Native structure-based models do, however, include repulsion between residues due to excluded volume. Nonenergetic traps can therefore exist if the chain, which cannot cross over itself, must partially unfold so that folding can proceed. The ability of native structure-based models to capture this kind of motion is partly responsible for their successful predictions of folding pathways for many types of proteins. Models without frustration describe the sequence of folding events for cytochrome c well (as inferred from hydrogen-exchange experiments), thereby justifying their use as a starting point. At low pH, the experimentally observed folding sequence of cytochrome c deviates from that at pH 7 and from models with perfectly funneled energy landscapes. Here, alternate folding pathways are a result of "chemical frustration". This frustration arises because some regions of the protein are destabilized more than others due to the heterogeneous distribution of titratable residues that are protonated at low pH. Beginning with native structure-based terms, we construct more complex models by adding chemical frustration. These more complex models only modestly perturb the energy landscape, which remains, overall, well funneled. These perturbed models can accurately describe how alternative folding pathways are used at low pH. At alkaline pH, cytochrome c populates distinctly different structural ensembles. For instance, lysine residues are deprotonated and compete for the heme ligation site. The same models that can describe folding at low pH also predict well the structures and relative stabilities of intermediates populated at alkaline pH. The success of models based on funneled energy landscapes suggest that cytochrome c folding is driven primarily by native contacts. The presence of heme appears to add chemical complexity to the folding process, but it does not require fundamental modification of the general principles used to describe folding. Moreover, its added complexity provides a valuable means of probing the folding energy landscape in greater detail than is possible with simpler systems.

Understanding protein folding cooperativity based on topological consideration

The folding cooperativity is an important issue of protein folding dynamics. Since the native topology plays a significant role in determining the folding behavior of proteins, we believe that it also has close relationship with the folding cooperativity. In the present work, we perform simulations on proteins Naf-BBL, QNND-BBL, CI2, and SH3 with the Gō model and compare their different folding behaviors. By analyzing the weak cooperative folding of protein Naf-BBL in detail, we found that the folding of Naf-BBL shows relatively weak thermodynamic coupling between residues, and such weak coupling is found mainly between the nonlocal native contacts. This finding complements our understandings on the source of barrierless folding of Naf-BBL and promotes us to analyze the topological origins of the poor thermodynamic coupling of Naf-BBL. Then, we further extend our analysis to other two-state and multistate proteins. Based on the considerations of the thermodynamic coupling and kinetic coupling, we conclude that the fraction of scattered native contacts, the difference in loop entropy of contacts, and the long range relative contact order are the major topological factors that influence the folding cooperativity. The combination of these three tertiary structural features shows significant correlations with the folding types of proteins. Moreover, we also discuss the topological factors related to downhill folding. Finally, the generic role of tertiary structure in determining the folding cooperativity is summarized.

How effective for fold recognition is a potential of mean force that includes relative orientations between contacting residues in proteins?

We estimate the statistical distribution of relative orientations between contacting residues from a database of protein structures and evaluate the potential of mean force for relative orientations between contacting residues. Polar angles and Euler angles are used to specify two degrees of directional freedom and three degrees of rotational freedom for the orientation of one residue relative to another in contacting residues, respectively. A local coordinate system affixed to each residue based only on main chain atoms is defined for fold recognition. The number of contacting residue pairs in the database will severely limit the resolution of the statistical distribution of relative orientations, if it is estimated by dividing space into cells and counting samples observed in each cell. To overcome such problems and to evaluate the fully anisotropic distributions of relative orientations as a function of polar and Euler angles, we choose a method in which the observed distribution is represented as a sum of delta functions each of which represents the observed orientation of a contacting residue, and is evaluated as a series expansion of spherical harmonics functions. The sample size limits the frequencies of modes whose expansion coefficients can be reliably estimated. High frequency modes are statistically less reliable than low frequency modes. Each expansion coefficient is separately corrected for the sample size according to suggestions from a Bayesian statistical analysis. As a result, many expansion terms can be utilized to evaluate orientational distributions. Also, unlike other orientational potentials, the uniform distribution is used for a reference distribution in evaluating a potential of mean force for each type of contacting residue pair from its orientational distribution, so that residue-residue orientations can be fully evaluated. It is shown by using decoy sets that the discrimination power of the orientational potential in fold recognition increases by taking account of the Euler angle dependencies and becomes comparable to that of a simple contact potential, and that the total energy potential taken as a simple sum of contact, orientation, and (phi,psi) potentials performs well to identify the native folds.

In quest of an empirical potential for protein structure prediction

Key to successful protein structure prediction is a potential that recognizes the native state from misfolded structures. Recent advances in empirical potentials based on known protein structures include improved reference states for assessing random interactions, sidechain-orientation-dependent pair potentials, potentials for describing secondary or supersecondary structural preferences and, most importantly, optimization protocols that sculpt the energy landscape to enhance the correlation between native-like features and the energy. Improved clustering algorithms that select native-like structures on the basis of cluster density also resulted in greater prediction accuracy. For template-based modeling, these advances allowed improvement in predicted structures relative to their initial template alignments over a wide range of target-template homology. This represents significant progress and suggests applications to proteome-scale structure prediction.

Energy barriers, cooperativity, and hidden intermediates in the folding of small proteins

Current theoretical views of the folding process of small proteins (< approximately 100 amino acids) postulate that the landscape of potential mean force (PMF) for the formation of the native state has a funnel shape and that the free energy barrier to folding arises from the chain configurational entropy only. However, recent theoretical studies on the formation of hydrophobic clusters with explicit water suggest that a barrier should exist on the PMF of folding, consistent with the fact that protein folding generally involves a large positive activation enthalpy at room temperature. In addition, high-resolution structural studies of the hidden partially unfolded intermediates have revealed the existence of non-native interactions, suggesting that the correction of the non-native interactions during folding should also lead to barriers on PMF. To explore the effect of a PMF barrier on the folding behavior of proteins, we modified Zwanzig's model for protein folding with an uphill landscape of PMF for the formation of transition states. We found that the modified model for short peptide segments can satisfy the thermodynamic and kinetic criteria for an apparently two-state folding. Since the Levinthal paradox can be solved by a stepwise folding of short peptide segments, a landscape of PMF with a locally uphill search for the transition state and cooperative stabilization of folding intermediates/native state is able to explain the available experimental results for small proteins. We speculate that the existence of cooperative hidden folding intermediates in small proteins could be the consequence of the highly specific structures of the native state, which are selected by evolution to perform specific functions and fold in a biologically meaningful time scale.

A funneled energy landscape for cytochrome c directly predicts the sequential folding route inferred from hydrogen exchange experiments

Proteins fold through a variety of mechanisms. For a given protein, folding routes largely depend on the protein's stability and its native-state geometry, because the landscape is funneled. These ideas are corroborated for cytochrome c by using a coarse-grained topology-based model with a perfect funnel landscape that includes explicit modeling of the heme. The results show the importance of the heme as a nucleation site and explain the observed hydrogen exchange patterns of cytochrome c within the context of energy landscape theory.

Theoretical model of prion propagation: a misfolded protein induces misfolding

There is a hypothesis that dangerous diseases such as bovine spongiform encephalopathy, Creutzfeldt-Jakob, Alzheimer's, fatal familial insomnia, and several others are induced by propagation of wrong or misfolded conformations of some vital proteins. If for some reason the misfolded conformations were acquired by many such protein molecules it might lead to a "conformational" disease of the organism. Here, a theoretical model of the molecular mechanism of such a conformational disease is proposed, in which a metastable (or misfolded) form of a protein induces a similar misfolding of another protein molecule (conformational autocatalysis). First, a number of amino acid sequences composed of 32 aa have been designed that fold rapidly into a well defined native-like alpha-helical conformation. From a large number of such sequences a subset of 14 had a specific feature of their energy landscape, a well defined local energy minimum (higher than the global minimum for the alpha-helical fold) corresponding to beta-type structure. Only one of these 14 sequences exhibited a strong autocatalytic tendency to form a beta-sheet dimer capable of further propagation of protofibril-like structure. Simulations were done by using a reduced, although of high resolution, protein model and the replica exchange Monte Carlo sampling procedure.

Protein–protein interactions: organization, cooperativity and mapping in a bottom-up Systems Biology approach

Understanding and ultimately predicting protein associations is immensely important for functional genomics and drug design. Here, we propose that binding sites have preferred organizations. First, the hot spots cluster within densely packed 'hot regions'. Within these regions, they form networks of interactions. Thus, hot spots located within a hot region contribute cooperatively to the stability of the complex. However, the contributions of separate, independent hot regions are additive. Moreover, hot spots are often already pre-organized in the unbound (free) protein states. Describing a binding site through independent local hot regions has implications for binding site definition, design and parametrization for prediction. The compactness and cooperativity emphasize the similarity between binding and folding. This proposition is grounded in computation and experiment. It explains why summation of the interactions may over-estimate the stability of the complex. Furthermore, statistically, charge-charge coupling of the hot spots is disfavored. However, since within the highly packed regions the solvent is screened, the electrostatic contributions are strengthened. Thus, we propose a new description of protein binding sites: a site consists of (one or a few) self-contained cooperative regions. Since the residue hot spots are those conserved by evolution, proteins binding multiple partners at the same sites are expected to use all or some combination of these regions.

Conformational transition free energy profiles of an adsorbed, lattice model protein by multicanonical Monte Carlo simulation

Proteins often undergo changes in internal conformation upon interacting with a surface. We investigate the thermodynamics of surface induced conformational change in a lattice model protein using a multicanonical Monte Carlo method. The protein is a linear heteropolymer of 27 segments (of types A and B) confined to a cubic lattice. The segmental order and nearest neighbor contact energies are chosen to yield, in the absence of an adsorbing surface, a unique 3x3x3 folded structure. The surface is a plane of sites interacting either equally with A and B segments (equal affinity surface) or more strongly with the A segments (A affinity surface). We use a multicanonical Monte Carlo algorithm, with configuration bias and jump walking moves, featuring an iteratively updated sampling function that converges to the reciprocal of the density of states 1/Omega(E), E being the potential energy. We find inflection points in the configurational entropy, S(E)=k ln Omega(E), for all but a strongly adsorbing equal affinity surface, indicating the presence of free energy barriers to transition. When protein-surface interactions are weak, the free energy profiles F(E)=E-TS(E) qualitatively resemble those of a protein in the absence of a surface: a free energy barrier separates a folded, lowest energy state from globular, higher energy states. The surface acts in this case to stabilize the globular states relative to the folded state. When the protein surface interactions are stronger, the situation differs markedly: the folded state no longer occurs at the lowest energy and free energy barriers may be absent altogether.

Finite size effects on thermal denaturation of globular proteins

Finite size effects on the cooperative thermal denaturation of proteins are considered. A dimensionless measure of cooperativity, Omegac, scales as Nzeta, where N is the number of amino acids. Surprisingly, we find that zeta is universal with zeta=1+gamma, where the exponent gamma characterizes the divergence of the susceptibility for a self-avoiding walk. Our lattice model simulations and experimental data are consistent with the theory. Our finding rationalizes the marginal stability of proteins and substantiates the earlier predictions that the efficient folding of two-state proteins requires TF approximately Ttheta, where Ttheta and TF are the collapse and folding transition temperatures, respectively.

Generalized-ensemble algorithms: enhanced sampling techniques for Monte Carlo and molecular dynamics simulations

In complex systems with many degrees of freedom such as spin glass and biomolecular systems, conventional simulations in canonical ensemble suffer from the quasi-ergodicity problem. A simulation in generalized ensemble performs a random walk in potential energy space and overcomes this difficulty. From only one simulation run, one can obtain canonical ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review the generalized ensemble algorithms. Three well-known methods, namely, multicanonical algorithm (MUCA), simulated tempering (ST), and replica-exchange method (REM), are described first. Both Monte Carlo (MC) and molecular dynamics (MD) versions of the algorithms are given. We then present five new generalized-ensemble algorithms which are extensions of the above methods.

Three-body interactions improve the prediction of rate and mechanism in protein folding models

Here we study the effects of many-body interactions on rate and mechanism in protein folding by using the results of molecular dynamics simulations on numerous coarse-grained Calpha-model single-domain proteins. After adding three-body interactions explicitly as a perturbation to a Gō-like Hamiltonian with native pairwise interactions only, we have found (i) a significantly increased correlation with experimental phi values and folding rates, (ii) a stronger correlation of folding rate with contact order, matching the experimental range in rates when the fraction of three-body energy in the native state is approximately 20%, and (iii) a considerably larger amount of three-body energy present in chymotripsin inhibitor than in the other proteins studied.

Simple two-state protein folding kinetics requires near-levinthal thermodynamic cooperativity

Simple two-state folding kinetics of many small single-domain proteins are characterized by chevron plots with linear folding and unfolding arms consistent with an apparent two-state description of equilibrium thermodynamics. This phenomenon is hereby recognized as a nontrivial heteropolymer property capable of providing fundamental insight into protein energetics. Many current protein chain models, including common lattice and continuum Gō models with explicit native biases, fail to reproduce this generic protein property. Here we show that simple two-state kinetics is obtainable from models with a cooperative interplay between core burial and local conformational propensities or an extra strongly favorable energy for the native structure. These predictions suggest that intramolecular recognition in real two-state proteins is more specific than that envisioned by common Gō-like constructs with pairwise additive energies. The many-body interactions in the present kinetically two-state models lead to high thermodynamic cooperativity as measured by their van't Hoff to calorimetric enthalpy ratios, implying that the native and denatured conformational populations are well separated in enthalpy by a high free-energy barrier. It has been observed experimentally that deviations from Arrhenius behavior are often more severe for folding than for unfolding. This asymmetry may be rationalized by one of the present modeling scenarios if the effective many-body cooperative interactions stabilizing the native structure against unfolding is less dependent on temperature than the interactions that drive the folding kinetics.

Contact order dependent protein folding rates: kinetic consequences of a cooperative interplay between favorable nonlocal interactions and local conformational preferences

Physical mechanisms underlying the empirical correlation between relative contact order (CO) and folding rate among naturally occurring small single-domain proteins are investigated by evaluating postulated interaction schemes for a set of three-dimensional 27mer lattice protein models with 97 different CO values. Many-body interactions are constructed such that contact energies become more favorable when short chain segments sequentially adjacent to the contacting residues adopt native-like conformations. At a given interaction strength, this scheme leads to folding rates that are logarithmically well correlated with CO (correlation coefficient r = 0.914) and span more than 2.5 orders of magnitude, whereas folding rates of the corresponding Gō models with additive contact energies have much less logarithmic correlation with CO and span only approximately one order of magnitude. The present protein chain models also exhibit calorimetric cooperativity and linear chevron plots similar to that observed experimentally for proteins with apparent simple two-state folding/unfolding kinetics. Thus, our findings suggest that CO-dependent folding rates of real proteins may arise partly from a significant positive coupling between nonlocal contact favorabilities and local conformational preferences.

A simple lattice model that exhibits a protein-like cooperative all-or-none folding transition

In a recent paper (D. Gront et al., Journal of Chemical Physics, Vol. 115, pp. 1569, 2001) we applied a simple combination of the Replica Exchange Monte Carlo and the Histogram methods in the computational studies of a simplified protein lattice model containing hydrophobic and polar units and sequence-dependent local stiffness. A well-defined, relatively complex Greek-key topology, ground (native) conformations was found; however, the cooperativity of the folding transition was very low. Here we describe a modified minimal model of the same Greek-key motif for which the folding transition is very cooperative and has all the features of the "all-or-none" transition typical of real globular proteins. It is demonstrated that the all-or-none transition arises from the interplay between local stiffness and properly defined tertiary interactions. The tertiary interactions are directional, mimicking the packing preferences seen in proteins. The model properties are compared with other minimal protein-like models, and we argue that the model presented here captures essential physics of protein folding (structurally well-defined protein-like native conformation and cooperative all-or-none folding transition).

A minimal physically realistic protein-like lattice model: designing an energy landscape that ensures all-or-none folding to a unique native state

A simple protein model restricted to the face-centered cubic lattice has been studied. The model interaction scheme includes attractive interactions between hydrophobic (H) residues, repulsive interactions between hydrophobic and polar (P) residues, and orientation-dependent P-P interactions. Additionally, there is a potential that favors extended beta-type conformations. A sequence has been designed that adopts a native structure, consisting of an antiparallel, six-member Greek-key beta-barrel with protein-like structural degeneracy. It has been shown that the proposed model is a minimal one, i.e., all the above listed types of interactions are necessary for cooperative (all-or-none) type folding to the native state. Simulations were performed via the Replica Exchange Monte Carlo method and the numerical data analyzed via a multihistogram method.

Efficiency of the multicanonical simulation method as applied to peptides of increasing size: the heptapeptide deltorphin

The advantage of the multicanonical (MUCA) simulation method of Berg and coworkers over the conventional Metropolis method is in its ability to move a system effectively across energy barriers thereby providing results for a wide range of temperatures. However, a MUCA simulation is based on weights (related to the density of states) that should be determined prior to a production run and their calculation is not straightforward. To overcome this difficulty a procedure has been developed by Berg that calculates the MUCA weights automatically. In a previous article (Yaşar et al. J Comput Chem 2000, 14, 1251-1261) we extended this procedure to continuous systems and applied it successfully to the small pentapeptide Leu-enkephalin. To investigate the performance of the automated MUCA procedure for larger peptides, we apply it here to deltorphin, a linear heptapeptide with bulky side chains (H-Tyr(1)-D-Met(2)-Phe(3)-His(4)-Leu(5)-Met(6)-Asp(7)-NH(2)). As for Leu-enkephalin, deltorphin is modeled in vacuum by the potential energy function ECEPP. MUCA is found to perform well. A weak second peak is seen for the specific heat, which is given a special attention. By minimizing the energy of structures along the trajectory it is found that MUCA provides a good conformational coverage of the low energy region of the molecule. These latter results are compared with conformational coverage obtained by the Monte Carlo minimization method of Li and Scheraga.

Anti-cooperativity and cooperativity in hydrophobic interactions: Three-body free energy landscapes and comparison with implicit-solvent potential functions for proteins

Potentials of mean force (PMFs) of three-body hydrophobic association are investigated to gain insight into similar processes in protein folding. Free energy landscapes obtained from explicit simulations of three methanes in water are compared with that predicted by popular implicit-solvent effective potentials for the study of proteins. Explicit-water simulations show that for an extended range of three-methane configurations, hydrophobic association at 25 degrees C under atmospheric pressure is mostly anti-cooperative, that is, less favorable than if the interaction free energies were pairwise additive. Effects of free energy nonadditivity on the kinetic path of association and the temperature dependence of additivity are explored by using a three-methane system and simplified chain models. The prevalence of anti-cooperativity under ambient conditions suggests that driving forces other than hydrophobicity also play critical roles in protein thermodynamic cooperativity. We evaluate the effectiveness of several implicit-solvent potentials in mimicking explicit water simulated three-body PMFs. The favorability of the contact free energy minimum is found to be drastically overestimated by solvent accessible surface area (SASA). Both the SASA and a volume-based Gaussian solvent exclusion model fail to predict the desolvation barrier. However, this barrier is qualitatively captured by the molecular surface area model and a recent "hydrophobic force field." None of the implicit-solvent models tested are accurate for the entire range of three-methane configurations and several other thermodynamic signatures considered.

Thermodynamics and folding kinetics analysis of the SH3 domain form discrete molecular dynamics

We perform a detailed analysis of the thermodynamics and folding kinetics of the SH3 domain fold with discrete molecular dynamic simulations. We propose a protein model that reproduces some of the experimentally observed thermodynamic and folding kinetic properties of proteins. Specifically, we use our model to study the transition state ensemble of the SH3 fold family of proteins, a set of unstable conformations that fold to the protein native state with probability 1/2. We analyze the participation of each secondary structure element formed at the transition state ensemble. We also identify the folding nucleus of the SH3 fold and test extensively its importance for folding kinetics. We predict that a set of amino acid contacts between the RT-loop and the distal hairpin are the critical folding nucleus of the SH3 fold and propose a hypothesis that explains this result.

Towards a consistent modeling of protein thermodynamic and kinetic cooperativity: how applicable is the transition state picture to folding and unfolding?

To what extent do general features of folding/unfolding kinetics of small globular proteins follow from their thermodynamic properties? To address this question, we investigate a new simplified protein chain model that embodies a cooperative interplay between local conformational preferences and hydrophobic burial. The present four-helix-bundle 55mer model exhibits protein-like calorimetric two-state cooperativity. It rationalizes native-state hydrogen exchange observations. Our analysis indicates that a coherent, self-consistent physical account of both the thermodynamic and kinetic properties of the model leads naturally to the concept of a native state ensemble that encompasses considerable conformational fluctuations. Such a multiple-conformation native state is seen to involve conformational states similar to those revealed by native-state hydrogen exchange. Many of these conformational states are predicted to lie below native baselines commonly used in interpreting calorimetric data. Folding and unfolding kinetics are studied under a range of intrachain interaction strengths as in experimental chevron plots. Kinetically determined transition midpoints match well with their thermodynamic counterparts. Kinetic relaxations are found to be essentially single-exponential over an extended range of model interaction strengths. This includes the entire unfolding regime and a significant part of a folding regime with a chevron rollover, as has been observed for real proteins that fold with non-two-state kinetics. The transition state picture of protein folding and unfolding is evaluated by comparing thermodynamic free energy profiles with actual kinetic rates. These analyses suggest that some chevron rollovers may arise from an internal frictional effect that increasingly impedes chain motions with more native conditions, rather than being caused by discrete deadtime folding intermediates or shifts of the transition state peak as previously posited.

Is there a unique melting temperature for two-state proteins?

Thermal unfolding (or folding) in many proteins occurs in an apparent two-state manner, suggesting that only two states, unfolded and folded, are populated. At the melting temperature, Tm, the two states coexist. Using lattice models with side chains we show that individual residues become structured at temperatures that deviate from Tm, which implies that partially folded conformations make substantial contribution to thermodynamic properties of two-state proteins. We also find that the folding cooperativity for a given residue is linked to its accessible surface area. These results are consistent with the experiments on GCN4-like zipper peptide, which showed that local melting temperatures differ from Tm. Analysis of thermal unfolding of six proteins shows that deltaT/Tm approximately N(-1), where deltaT is the transition width and N is the number of residues. This scaling allows us to conclude that, when corrected for finite size effects, folding cooperativity can be captured using coarse grained models.

Generalized-ensemble algorithms for molecular simulations of biopolymers

In complex systems with many degrees of freedom such as peptides and proteins, there exists a huge number of local-minimum-energy states. Conventional simulations in the canonical ensemble are of little use, because they tend to get trapped in states of these energy local minima. A simulation in generalized ensemble performs a random walk in potential energy space and can overcome this difficulty. From only one simulation run, one can obtain canonical-ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review uses of the generalized-ensemble algorithms in biomolecular systems. Three well-known methods, namely, multicanonical algorithm, simulated tempering, and replica-exchange method, are described first. Both Monte Carlo and molecular dynamics versions of the algorithms are given. We then present three new generalized-ensemble algorithms that combine the merits of the above methods. The effectiveness of the methods for molecular simulations in the protein folding problem is tested with short peptide systems.

Matching simulation and experiment: a new simplified model for simulating protein folding

Simulations of simplified protein folding models have provided much insight into solving the protein folding problem. We propose here a new off-lattice bead model, capable of simulating several different fold classes of small proteins. We present the sequence for an alpha/beta protein resembling the IgG-binding proteins L and G. The thermodynamics of the folding process for this model are characterized using the multiple multihistogram method combined with constant-temperature Langevin simulations. The folding is shown to be highly cooperative, with chain collapse nearly accompanying folding. Two parallel folding pathways are shown to exist on the folding free energy landscape. One pathway contains an intermediate--similar to experiments on protein G, and one pathway contains no intermediates-similar to experiments on protein L. The folding kinetics are characterized by tabulating mean-first passage times, and we show that the onset of glasslike kinetics occurs at much lower temperatures than the folding temperature. This model is expected to be useful in many future contexts: investigating questions of the role of local versus nonlocal interactions in various fold classes, addressing the effect of sequence mutations affecting secondary structure propensities, and providing a computationally feasible model for studying the role of solvation forces in protein folding.

Energetic components of cooperative protein folding

A new lattice protein model with a four-helix bundle ground state is analyzed by a parameter-space Monte Carlo histogram technique to evaluate the effects of an extensive variety of model potentials on folding thermodynamics. Cooperative helical formation and contact energies based on a 5-letter alphabet are found to be insufficient to satisfy calorimetric and other experimental criteria for two-state folding. Such proteinlike behaviors are predicted, however, by models with polypeptidelike local conformational restrictions and environment-dependent hydrogen-bondinglike interactions.